In (I), we use \(d_1(\rho _{\widetilde{\Theta },\epsilon },\rho _{\Theta ,\epsilon })=O(\epsilon )\)  and a property of Fréchet derivative ([1]}) to obtain that
\(\left|E_s(\rho _{\widetilde{\Theta },\epsilon })-E_s(\rho _{\Theta })-\left[\int ^1_0\int _{\mathbb {R}^k}\frac{\delta E_s(\rho )}{\delta \rho _\Theta }(\theta ,t)\left(\,\mathrm {d}\rho _{\widetilde{\Theta },\epsilon }-\,\mathrm {d}\rho _{\Theta }\right)\,\mathrm {d}t\right]\right|\le o(d_1(\rho _{\widetilde{\Theta },\epsilon },\rho _{\Theta ,\epsilon }))=o(\epsilon )\,.\) 
